Abstract

A simple treatment by scalar-wave theory yields upper bounds to the efficiency of nonimaging concentrators that are lower than those given by geometrical optics.

© 1982 Optical Society of America

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References

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  1. W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).
  2. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).
    [Crossref]
  3. A. Walther, “Propagation of the generalized radiance through lenses,” J. Opt. Soc. Am. 68, 1606–1610 (1978); T. Jannson, “Radiance transfer function,” J. Opt. Soc. Am. 70, 1544–1549 (1980). Both papers discuss propagation of B through image-forming systems, but the treatment is restricted to systems that are isoplanatic and to radiance distributions from quasi-homogeneous sources; in our case neither of these restrictions applies (although the entry aperture would be in effect a quasi-homogeneous source if illuminated by a distant Lambertian source of finite angular extent, this would not be so for the plane of the exit aperture and intermediate planes). Thus, although Walther3 speaks of a “radiance invariance theorem” (i.e., radiance is invariant along rays), this deals only with cases in which well-defined rays can be defined and traced; but the ray concept becomes meaningless close to diffracting edges, as in Fig. 1, and we are interested specifically in cases in which such diffraction occurs.
    [Crossref]
  4. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

1978 (1)

1973 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Walther, A.

Welford, W. T.

W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).

Winston, R.

W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

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Figures (5)

Fig. 1
Fig. 1

A compound parabolic concentrator of concentration ratio 2. Rays at the extreme entry angle are shown.

Fig. 2
Fig. 2

An ideal image-forming system as a concentrator. (The profile of the lens is not intended to be realistic.)

Fig. 3
Fig. 3

Images of a source of unit intensity and angular subtenses corresponding to the values of z0 shown. The areas under the curves from the origin to z0 show the efficiency at the design concentration ratio.

Fig. 4
Fig. 4

Efficiency η as a function of z1, the half-width of the exit aperture for different values of z0.

Fig. 5
Fig. 5

Ratio of concentration to the design value as a function of efficiency.

Equations (14)

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H = B ( r , n ) cos θ d Ω d 2 r .
h = a sin β / sin α ,
z 1 = k h 1 sin α ,
z 1 = k a sin β 1 ,
2 sin α λ δ β W ( sin z 1 z 1 ) 2 .
2 W sin α f λ - f sin β 0 f sin β 0 [ sin ( z - z 1 ) z - z 1 ] d h 1 ,
W sin α a · ½ [ G ( z 0 + z ) + sgn ( z 0 - z ) G ( z 0 - z ) ] ,
G ( x ) = 2 π [ Si ( 2 x ) - sin 2 x x ]             ( x 0 ) .
G ( x ) ~ 1 - 1 π ( 1 x + sin 2 x 2 x 2 ) .
2 W k a 0 z 1 I ( z ; z 0 ) d z ,
η = 1 z 0 0 z 1 I ( z ; z 0 ) d z .
C g = a h 1 = z 0 z 1 sin α sin β 0 ,
C g = a k sin α z 1 .
W π f z - z 0 z + z 0 ( sin t t ) 2 d t .